The 70-300 has generally better IQ towards the edge of the frame IMO and with USM, it focuses more quickly too. IF you see yourself moving to a full-frame body, the 70-300 will still be compatible whereas the 55-250 will not.

Don't get me wrong, I'm not writing off the 55-250 as it's a competent lens but for the extra money, you arguably get a better lens with the 70-300.

If you decide to go for the 70-300, you may also want to consider the Tamron 70-300mm as it's reportedly even sharper than Canon's version.

For the money, the 55-250 IS is a very good lens (as well as it's light-weight). Of course when spending more money you'll be getting better and sharper lenses, but then Id rather look at the mentioned Tamron instead!

+1 for the Tamron. It's full frame compatible as well. In my opinion the full frame compatibility is getting more and more relevant as the old full frame cameras are getting into a realistic price range. (EOS 5D ~ 650€ / EOS 1D2 Mark II ~ 1000€)